Definition:General Logarithm/Common

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Definition

Logarithms base $10$ are often referred to as common logarithms.


Notation for Negative Logarithm

Let $n \in \R$ be a real number such that $0 < n < 1$.

Let $n$ be presented (possibly approximated) in scientific notation as:

$a \times 10^{-d}$

where $d \in \Z_{>0}$ is a (strictly) positive integer.

Let $\log_{10} n$ denote the common logarithm of $n$.

Then it is the standard convention to express $\log_{10} n$ in the form:

$\log_{10} n = \overline d \cdotp m$

where $m := \log_{10} a$ is the mantissa of $\log_{10} n$.


The overline notation is commonly read as bar, that is:

$\overline 2$ is read as bar two.


Mantissa

Let $\log_{10} n$ be expressed in the form:

$\log_{10} n = \begin {cases} c \cdotp m & : d \ge 0 \\ \overline c \cdotp m & : d < 0 \end {cases}$

where:

$c = \size d$ is the absolute value of $d$
$m := \log_{10} a$


$\log_{10} a$ is the mantissa of $\log_{10} n$.


Characteristic

$c$ is the characteristic of $\log_{10} n$.


Examples

Common Logarithm: $\log_{10} 1000$

The common logarithm of $1000$ is:

$\log_{10} 1000 = 3$


Common Logarithm: $\log_{10} 0 \cdotp 01$

The common logarithm of $0 \cdotp 01$ is:

$\log_{10} 0 \cdotp 01 = -2$


Common Logarithm: $\log_{10} 2$

The common logarithm of $2$ is:

$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$


Common Logarithm: $\log_{10} 3$

The common logarithm of $3$ is:

$\log_{10} 3 = 0.47712 \, 12547 \, 19662 \, 43729 \, 50279 \ldots$


Common Logarithm: $\log_{10} e$

The common logarithm of Euler's number $e$ is:

$\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03251 \, 82765 \, 11289 \, 18916 \, 60508 \, 22943 \, 97005 \, 803 \ldots$


Common Logarithm: $\log_{10} \pi$

The common logarithm of $\pi$ is:

$\log_{10} \pi = 0.49714 \, 98726 \, 94133 \, 85435 \, 12683 \ldots$


Also known as

Common logarithms are sometimes referred to as Briggsian logarithms or Briggs's logarithms, for Henry Briggs.


In elementary textbooks and on most pocket calculators, $\log$ is assumed to mean $\log_{10}$.

This ambiguous notation is not recommended, particularly since $\log$ often means base $e$ in more advanced textbooks.


Also see

  • Results about logarithms can be found here.


Historical Note

Common logarithms were developed by Henry Briggs, as a direct offshoot of the work of John Napier.


After seeing the tables that Napier published, Briggs consulted Napier, and suggested defining them differently, using base $10$.

In $1617$, Briggs published a set of tables of logarithms of the first $1000$ positive integers.

In $1624$, he published tables of logarithms which included $30 \, 000$ logarithms going up to $14$ decimal places.


Before the advent of cheap means of electronic calculation, common logarithms were widely used as a technique for performing multiplication.


Linguistic Note

The word logarithm comes from the Ancient Greek λόγος (lógos), meaning word or reason, and ἀριθμός (arithmós), meaning number.


Sources

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